Conic singularities metrics with prescribed Ricci curvature: General cone angles along normal crossing divisors

Mihai Päun (1) , Henri Guenancia (2)

1 KIAS Séoul - School of Mathematics
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

Let X be a non-singular compact Kähler manifold, endowed with an effective divisor D = (1 − β k)Y k having simple normal crossing support, and satisfying β k ∈ (0, 1). The natural objects one has to consider in order to explore the differential-geometric properties of the pair (X, D) are the so-called metrics with conic singularities. In this article, we complete our earlier work [CGP13] concerning the Monge-Ampère equations on (X, D) by establishing Laplacian and C 2,α,β estimates for the solution of this equations regardless to the size of the coefficients 0 < β k < 1. In particular, we obtain a general theorem concerning the existence and regularity of Kähler-Einstein metrics with conic singularities along a normal crossing divisor.

Mots clés: Kähler-Einstein metrics; conic singularities; orbifold tensors; Monge-Ampère equations; klt pairs

Domaines: Mathématiques [math]; Variables complexes [math.CV]; Géométrie différentielle [math.DG]

Mihai Päun, Henri Guenancia. Conic singularities metrics with prescribed Ricci curvature: General cone angles along normal crossing divisors. Journal of Differential Geometry, 2016, 103 (1), pp.15 - 57. ⟨10.4310/jdg/1460463562⟩. ⟨hal-01878996⟩